Vanessa works out for $\frac{3}{4}$ of an hour every day. To keep her exercise routines interesting, she includes different types of exercises, such as sit-ups and jumping jacks, in each workout. If each type of exercise takes $\frac{3}{20}$ of an hour, how many different types of exercise can Vanessa do in each workout?
Solution: To find out how many types of exercise Vanessa could do in each workout, divide the total amount of exercise time ( $\frac{3}{4}$ of an hour) by the amount of time each exercise type takes ( $\frac{3}{20}$ of an hour). $ \dfrac{{\dfrac{3}{4} \text{ hour}}} {{\dfrac{3}{20} \text{ hour per exercise}}} = {\text{ number of exercises}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{3}{20} \text{ hour per exercise}}$ is ${\dfrac{20}{3} \text{ exercises per hour}}$ $ {\dfrac{3}{4}\text{ hour}} \times {\dfrac{20}{3} \text{ exercises per hour}} = {\text{ number of exercises}} $ $ \dfrac{{3} \cdot {20}} {{4} \cdot {3}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $3$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{\cancel{3}^{1}} \cdot {20}} {{4} \cdot {\cancel{3}^{1}}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $20$ in the numerator and the $4$ in the denominator by $4$ $ \dfrac{{1} \cdot {\cancel{20}^{5}}} {{\cancel{4}^{1}} \cdot {1}} = {\text{ number of exercises}} $ Simplify: $ \dfrac{{1} \cdot {5}} {{1} \cdot {1}} = {5} $ Vanessa can do 5 different types of exercise per workout.